A model for cage formation in colloidal suspension of laponite

Yogesh M. Joshi

Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur

208016, INDIA.

E-Mail: joshi@iitk.ac.in

Telephone: +91 512 259 7993, Fax: +91 512 259 0104

Abstract

In this paper we investigate glass transition in aqueous suspension of

synthetic hectorite clay, laponite. We believe that upon dispersing laponite clay

in water, system comprises of clusters (agglomerates) of laponite dispersed in

the same. Subsequent osmotic swelling of these clusters leads to increase in

their volume fraction. We propose that this phenomenon is responsible for

slowing down of the overall dynamics of the system. As clusters fill up the

space, system undergoes glass transition. Along with the mode coupling theory,

proposed mechanism rightly captures various characteristic features of the

system in the ergodic regime as it approaches glass transition.

INTRODUCTION:

Glasses are disordered materials, and in such state, system explores only a

small part of the phase space. Molecular glasses are formed by quenching the liquid

rapidly below so called glass transition temperature to obtain a disordered structure.1

In colloidal suspensions, the glassy state is attained by increasing the concentration

of the constituent particles such that the disordered state is obtained above a random

loose packing threshold.2 Generally this is achieved by either centrifugation3 or

dialysis.4 Thus, colloidal glasses are different from molecular glasses wherein the

concentration, not the temperature, plays an important role in causing arrested or

glassy state. In this paper we investigate the kinetics of glass transition in colloidal

glass of aqueous laponite suspension wherein osmotic swelling of laponite clusters

brings about glass transition.

2

Lately, the colloidal glasses formed by aqueous suspensions of laponite have

developed considerable interest.5-10 Laponite particle has a disc-like shape, and due to

negative charges on its surface, repulsive interactions prevail among particles in low

ionic concentration aqueous medium that leads to a formation of the so called Wigner

glass.10 Apart from having wide ranging industrial applications; these suspensions

are considered as model systems to study slow dynamics of glass transition and

gelation.11

The laponite disc has diameter of 25 nm and layer thickness of 1 nm. At very

low ionic concentrations its suspension in ultra pure water leads to a glassy state for

volume fractions less than 0.01 in contrast to colloidal glasses of uncharged spherical

particles where glassy state is attained at volume fractions only above 0.5. This is

essentially due to anisotropic shape of a laponite disc and negatively charged surface.

Later gives rise to electrostatic screening length of 30 nm.12 Hence the effective

volume of a single laponite disc increases by a factor of 60 and explains the

requirement of approximately 60 times lower volume fraction for glass formation.

(Volume of single laponite particle is of the order of 1×25×25 nm3. However the

effective volume of laponite particle including electrostatic screening length of 30 nm

on both the sides becomes 60×25×25 nm3)

In the past few years several groups have studied liquid-glass transition of a

laponite suspension at low ionic concentration using various optical techniques.5-10,12-14

In general laponite powder is mixed in ultra-pure water with vigorous stirring and

passed through a micro-filter to study the evolution of its structure with respect to

age. The suspension shows two relaxation modes with the fast mode being

independent of age.13,14 Its inverse square dependence on wave vector ( q ) highlights

its diffusive character. The slow mode shows an initial rapid increase followed by a

linear increase with respect to age.5,15 The former regime is called the cage forming

regime,15 where the slow mode also shows inverse square dependence on q . In the full

aging regime, it shows q dependence with power law index in the range –1 to –1.3.

Furthermore, the age at which transition from rapid growth to linear growth occurs,

decreases with increase in clay concentration.5 Very recently Mossa et al.,16 while

studying ageing of laponite suspension in the nonergodic regime using Brownian

dynamics simulations, observed that anisotropy of the platelet along with repulsive

3

Yukawa potential is responsible for the local disorder that takes the system into the

metastable state. They further observed that ageing dynamics strongly affects

orientational degree of freedom which relaxes over the timescale of translational

modes. There are various other morphologies proposed in the literature, and in

general the state of the system in full ageing regime, gel or glass, still remains a

contentious issue.17

In the literature, the rapid growth of relaxation time with respect to age in the

ergodic regime is fitted by an empirical function ( )ln ~ wtτ and hence is generally

referred to as an exponential growth.5,13-15 Tanaka et al.15 proposed two-stage aging

kinetics by expressing the relaxation time by using the average barrier height U for

the particle motion as: ( )~ exp U kTτ . They suggested that the barrier height should

grow linearly with aging time in the ergodic regime while logarithmically in the full

aging regime to yield exponential and linear age dependence of relaxation time

respectively. However they mentioned that the basis for logarithmic dependence that

leads to exponential growth is not clear. They concluded by asserting that that rapid

increase in slow mode with age in the ergodic regime remains an unanswered

question. Schosseler et al.,5 after experimentally investigating this phenomenon,

linked it to liquid-glass transition, but did not conclude about the mechanism. In this

paper we investigate this very phenomenon. For the first time, we show that the

osmotic swelling of laponite clusters bring about rapid increase in the relaxation time

as the system approaches arrested state. We limit our discussion only to cage forming

regime.

THEORY:

We consider a system of aqueous suspension of laponite having basic pH that

leads to net negative charge on the laponite particle.18 The chemical formula for

laponite is Na+0.7[(Si8Mg5.5Li0.3)O20(OH)4]–0.7. Isomorphic substitution of magnesium by

lithium atoms generates negative charges on its surface that are counterbalanced by

the positive charge of the sodium ions present in the interlayer. We assume that

immediately after stirring/filtration the system comprises of clusters (agglomerates) of

laponite dispersed in water. In the present model, the word cluster should be referred

to as agglomerate that is composed of domains of parallel stacks of laponite particles.

4

As time progresses, the clusters of laponite undergo swelling due to diffusion (flux) of

water into the cluster driven by an osmotic pressure gradient that is caused by

repulsive interactions between neighboring particles.

We believe that the clusters of laponite get hydrated while preparing the

suspension by stirring. Filtration process breaks the clusters. For simplicity we

consider all the clusters to be spherical in shape, of same size and at the same level of

hydration. Let ψ be the volume fraction of laponite inside a cluster of radius 0R at

time wt =0. Just outside the cluster, the volume fraction of water is unity and imposes

an osmotic pressure gradient. As time progresses water diffuses into the cluster until

the concentration of water (or alternatively laponite, or the osmotic pressure) becomes

uniform over the space. If the overall volume fraction of laponite is φ , the number of

clusters per unit volume is given by,

( )30n Rφ ψ= . (1)

The clusters swell due to osmotic diffusion. If the ( )wR t is the radius of cluster at any

time wt , then the fraction of total volume occupied by clusters is 3( )wtξ φ ψ while the

mode coupling distance parameter ε is given by: 3( )c w ctε φ ξ φ ψ φ⎡ ⎤= −⎣ ⎦ , (2)

where 0( ) ( )w wt R t Rξ = is the dimensionless radius and cφ (~0.56-0.64) is the volume

fraction at which growing spherical clusters touch each other. We believe that as

0ε → ; the system undergoes a transition from a cage forming regime to a glassy or

full aging regime. The dimensionless radius at the onset of this transition can be

obtained from eq. (2) and is given by,

( )13

cξ φ ψ φ∗ = . (3)

In the powder form, laponite discs are present in the domains of parallel stacks

with sodium atoms present in the interlayer. Upon dispersing in water, swelling of

clay occurs in two steps. In the first step, water is absorbed in successive monolayers

on the surface that pushes the discs apart. As the hydration of clay takes place, the

Na+ ions, though are electrostatically attracted towards the oppositely charged

surface, diffuse away from the surface into the bulk where their concentration is low.

As some Na+ ions diffuse out of the gallery, the negatively charged surfaces of the

5

opposite faces get exposed to each other and hence repel. In the second stage of

swelling, double layer repulsion pushes the laponite discs further apart.18 The larger

the distance between the laponite discs, lower is the osmotic pressure required to hold

them in the same position. Recently Martin and coworkers19 proposed a scaling model

to estimate flux of water through concentrated sediment of laponite under an osmotic

pressure gradient. Using analytical solution of Poisson-Boltzmann equation for the

case where only counterions are present in the interlayer space,20 they argued that

the osmotic pressure varies as the inverse square of the plate separation distance in

the concentration regime of usual interest [namely, plate separation distance 0.5 nm

to 30 nm for the present system that corresponds to concentration of about 33 vol % to

1.6 vol %]. Their expression for dependence of osmotic pressure on clay concentration

gives excellent prediction of the experimental data. Since the osmotic pressure

gradient drives the flow of water through porous sediment, they used Darcy’s law to

predict flow rate. If P∆ is the pressure difference over the radius of the cluster, the

flux of water having viscosity µ is given by ( )pQ k A P Rµ= ∆ , where pk is a

dimensionless constant that depends on the porosity and characteristic features of the

cluster (porous object) and A is the surface area of the sphere. Considering an

incremental swelling step, where flux Q over time wdt causes an increase in volume

by dV , we get wQdt dV= .19 This leads to p wRdR Pk dtµ = ∆ . Based on the analytical

solution for Poisson-Boltzmann equation, Martin and coworkers19 argued that

dependence of product pk P∆ on volume fraction of laponite in the cluster (and hence

time) can be neglected in the concentration regime of interest. The above equation can

be easily integrated to predict the size of the swelling cluster under osmotic pressure

gradient with respect to time to yield,

( )1

220( ) 1 /w wt Dt Rξ ⎡ ⎤= +⎣ ⎦ . (4)

where wt is waiting time and D is characteristic diffusivity of the swelling process in

the limit of small volume fraction of laponite. It is given as, 2 2

02 2

2 4( )45

p r

l

Pk kT aDe t

π ε εµ µ

∆= ≈ ,

where k is Boltzmann constant, T is absolute temperature, 0ε is permittivity of

vacuum, rε is relative permittivity of water medium, e is unit charge, lt is thickness

6

of laponite plate and a is equivalent radius that gives the same mass per particle as a

laponite particle (0.82 nm).19 Thus the characteristic diffusivity can be readily

evaluated by knowing the temperature of the system.

Immediately after filtration, the system is filled with many clusters undergoing

Brownian motion. As the volume fraction of clusters, 3( )wtξ φ ψ goes on increasing,

the available space that is not occupied by clusters decreases and the dynamics of the

system slows down. This increases the corresponding characteristic time-scale of the

system. This process is akin to glass transition, where, as temperature of a glass

forming liquid is decreased, movement of the constituent molecules gets constrained.

If we apply this analogy to the present system, the growth of the cluster can be

considered equivalent to decreasing the temperature of the molecular glass formers.

Under such conditions we can estimate the characteristic relaxation time of the

system using mode coupling theory that describes the behavior of colloidal glasses

very well. The relaxation time predicted by the mode-coupling theory is given by:21

0γτ τ ε −= , (5)

where, 0τ and γ are the fitting parameters.

The characteristic relaxation time can be obtained by incorporating eqs. (2) and

(4) into eq. (5). The model has three fitting parameters, volume of laponite in the

cluster ( )30R ψ , 0τ and γ . The first parameter arises naturally in the formulation and

cannot be prescribed a priori. This is because the radius of cluster and the volume

fraction of laponite inside a cluster at time 0wt = are strongly dependent on the time

of stirring. The latter parameter 0τ is observed to be of order unity for the present

system. Eq. (4) can be incorporated in eq. (3) to predict the transition time at which

system enters a nonergodic or full aging regime and is given by:

( ) ( )22 3 1 3 2 30w ct R Dφ ψ φ∗ = . (6)

However, due to uncertainty in getting reproducible data,5 it is difficult to validate

this expression experimentally as it demands ( )30R ψ to be identical in every

experiment. Combining eqs (2) to (6), we get:

( )3 2

0 1 w w w wt t t tγ

τ τ−

∗ ∗⎡ ⎤= − <⎢ ⎥⎣ ⎦… . (7)

7

Thus, knowledge of 0τ and wt∗ determine nature of evolution of relaxation time for

given value of γ .

DISCUSSION:

In figure 1, the characteristic dimensionless time scale ( )0τ τ of the suspension

is plotted as a function of dimensionless age of the sample ( )w wt t∗ for 2 and 2.75 wt. %

suspension of laponite in water,5 while an inset shows a log-log plot of 0τ τ vs.

( )3 21 w wt t∗− . As shown in the figure, eq. (7) with γ =2.58 provides an excellent fit to the

experimental data. Since the experimental data is made dimensionless by the model

parameters that are obtained from the experiments (namely 0τ and wt∗ ), only a unique

value of γ can fit the given data set. Interestingly γ =2.58 is the same value for which

classic mode coupling theory predicts glass transition in the monodispersed spherical

particles.21 However, the real sample of laponite suspension is expected have size

distribution of clusters and hence a fit of γ =2.58 to the experimental data might be

coincidental. As the volume fraction of the clusters approaches cφ , the characteristic

time-scale approaches the age of the system and undergoes a transition from a cage

forming regime (ergodic) to a full aging regime (nonergodic). This might not be a sharp

transition as the glassy state is attained only after laponite particles occupy the

available space completely, thus highlighting role of the length scale probed in the

experiment. Consequences of this result are discussed later in the paper. In the full

aging regime, system shows a linear relationship between characteristic timescale and

age. In the present model, in order to keep analysis simple, we have considered

osmotic swelling of clusters having same initial radius. According to eq. (4),

consideration of polydispersity in the initial radius of cluster will lead to change in

polydispersity with age. We believe that simplistic model and assumptions proposed

in the present manuscript is the first step to theoretically investigate the ergodicity

breaking mechanism in this system. A quantitative prediction provided by the present

model is indeed encouraging in that respect.

The relaxation time obtained by eq. (7) is a slow mode representing the

characteristic time associated with the cage diffusion process. In many experiments

8

that observe the rapid increase in slow relaxation mode with respect to time employ

sub- micron size tracer particles to strengthen the signal.5 As clusters grow,

movement of these tracer particles gets confined to smaller and smaller region in

space. Although this does not significantly affect characteristic time associated with

rattling motion within the confined space (fast mode), time required to escape the cage

formed by neighboring clusters increases rapidly. However the dynamics of the tracers

is still diffusive in nature leading to 2q− dependence. As growing laponite clusters fill

the available space completely, cage diffusion becomes extremely sluggish and the

corresponding hyperdiffusive timescale scales as age.

The concept modeled in this paper is similar to one of the ideas proposed by

Tanaka et al.15 where they speculate that the system becomes nonergodic after the

growing clusters of aggregates fill up the space. However, the present model is very

different than that proposed by the same group,15 where they argue that the average

barrier height for particle motion grows linearly with age in the ergodic regime. They

estimate the conductivity of the laponite suspension with respect to its age. They

found that the conductivity increases almost linearly with the aging time in the

ergodic regime, but tends to saturate in the nonergodic regime. The increase in

conductivity reflects the reduction in the number of strongly bound counterions. Our

model is in agreement with this observation. As the cluster size increases, more and

more counterions diffuse away into the bulk increasing the conductivity. As the

clusters fill up the space, along with laponite particles, the concentration of

counterions also becomes uniform, leading to saturation in the conductivity.

The present model clearly distinguishes between the ergodic state and non-

ergodic state based on the physical structure that exists in these two states. In the

former state clusters or agglomerates of laponite platelets are present, while in the

later regime, single laponite particle is an independent entity. It is well known that

aqueous suspension of laponite, when it is in the nonergodic regime, undergoes

rejuvenation due to excessive deformation.22 However, according to proposed physical

picture, the model clearly states that, due to rejuvenation, the system cannot cross the

ergodic-nonergodic transition point and enter ergodic regime. This means that

exponential-like rapid increase in slow relaxation mode cannot be observed again.

This observation has significant implications in analyzing rheological behavior of

9

laponite suspensions. Furthermore eq. (6) predicts inverse relationship between

transition time and characteristic diffusivity. We have seen that temperature

dependence of characteristic diffusivity ( )2( ) rD kT ε µ∝ comes from three terms,

namely, ( )2kT , permittivity of water and viscosity of water. Viscosity of water can be

considered to depend on temperature as 0U

kTeµ µ= , while permittivity of water, though

explicit analytical expression for its dependence on temperature does not exist,23

decreases weakly compared to that of viscosity of water. Overall, transition time is

expected to show significant decrease with respect to temperature. Ramsay24 studied

effect of temperature on rheological properties of ageing laponite suspension and

observed pronounced increase in elastic modulus with age at higher temperature. This

observation matches very well with the prediction of the model.

Nicolai et al.25 carried out static light scattering experiments on the aqueous

laponite dispersions in the concentration range 0.025 wt % to 0.5 wt. %. They observe

that intensity of scattered light decays significantly in the initial five hours followed

by a very sluggish decay. Dynamic light scattering experiments on the same sample

after one day showed system to be still ergodic, as expected for such low concentration

of laponite and suggested presence of individual laponite particles or an incomplete

dispersion of the oligomers with a broad size distribution. An initial decay in the

intensity of scattered light that eventually leads to oligomers or individual laponite

particles can be very well explained by the present model. As various clusters grow in

size, water content in the same increases, which decreases relative difference in the

refractive index between water and the cluster, decreasing the intensity of scattered

light.

Schosseler et al.5 have discussed various features of ergodic to nonergodic

transition in great details. They observed that immediately after filtration, the

viscosity of the suspension as recorded by the diffusivity of the tracer particles is of

the order of few mPas. This observation further strengthens the assumption that

laponite is present in the form of tiny clusters immediately after filtration. Schosseler

et al.5 further observed that that the full aging behavior is first seen while

investigating large length scales in the aging suspension of laponite. In the present

paper we argue that transition to nonergodic regime occurs when growing clusters of

10

laponite touch each other. However, the space between the clusters when they touch

each other still contains low viscosity aqueous medium which is ergodic in nature.

Thus, the transition to full ageing regime will not be observable until the probed

length scale is larger than the space between the growing clusters. Model captures

this behavior very well. Thus proposed model rightly captures various experimental

observations in the ergodic regime of aqueous suspension of laponite.

CONCLUSION:

We have modeled a new mode of glass transition in which clusters of laponite

particles undergo osmotic swelling and enter the non-ergodic state as they span the

available space. As the clusters fill up the space, cage diffusion process becomes very

sluggish. The mode coupling formalism along with proposed mechanism provides an

excellent prediction of the associated relaxation time dependence on age. Model also

predicts that the ergodic to nonergodic transition is first observed at large length

scales and it occurs at early age for higher temperature. These predictions are in

agreement with experimental observations.

ACKNOWLEDGEMENT:

Financial support from Department of Atomic Energy, Government of India

under the BRNS young scientist award scheme is greatly acknowledged. I would like

to thank Dr. Ranjini Bandyopadhyay, Dr. S. A. Ramakrishna and Dr. G.

Kumaraswamy for constructive remarks and discussion.

~~~~~~~~~~ 1 P. G. Debenedetti and F. H. Stillinger, Nature 410, 259 (2001). 2 K. J. Dong, R. Y. Yang, R. P. Zou, and A. B. Yu, Phys. Rev. Lett. 96, 145505 (2006). 3 P. N. Pusey and W. Van Megen, Phys. Rev. Lett. 59, 2083 (1987). 4 V. Viasnoff and F. Lequeux, Phys. Rev. Lett. 89, 065701 (2002). 5 F. Schosseler, S. Kaloun, M. Skouri, and J. P. Munch, Phys. Rev. E 73, 021401

(2006). 6 S. Bhatia, J. Barker, and A. Mourchid, Langmuir 19, 532 (2003). 7 B. Ruzicka, L. Zulian, and G. Ruocco, Phys. Rev. Lett. 93, 258301 (2004).

11

8 R. Bandyopadhyay, D. Liang, H. Yardimci, D. A. Sessoms, M. A. Borthwick, S. G. J.

Mochrie, J. L. Harden, and R. L. Leheny, Phys. Rev. Lett. 93, 228302 (2004). 9 P. Levitz, E. Lecolier, A. Mourchid, A. Delville, and S. Lyonnard, Europhys. Lett.

49, 672 (2000). 10 D. Bonn, H. Tanaka, G. Wegdam, H. Kellay, and J. Meunier, Europhys. Lett. 45, 52

(1998). 11 F. Sciortino and P. Tartaglia, Adv. Phys. 54, 471 (2005). 12 D. Bonn, H. Kellay, H. Tanaka, G. Wegdam, and J. Meunier, Langmuir 15, 7534

(1999). 13 B. Abou, D. Bonn, and J. Meunier, Phys. Rev. E 64, 215101 (2001). 14 M. Bellour, A. Knaebel, J. L. Harden, F. Lequeux, and J.-P. Munch, Phys. Rev. E

67, 031405 (2003). 15 H. Tanaka, S. Jabbari-Farouji, J. Meunier, and D. Bonn, Phys. Rev. E 71, 021402

(2005). 16 S. Mossa, C. De Michele, and F. Sciortino, J. Chem. Phys. 126, 014905 (2007). 17 P. Mongondry, J. F. Tassin, and T. Nicolai, J. Colloid and Interface Sci. 283, 397

(2005). 18 H. Van Olphen, An Introduction to Clay Colloid Chemistry. (Wiley New York, 1977). 19 C. Martin, F. Pignon, A. Magnin, M. Meireles, V. Lelievre, P. Lindner, and B.

Cabane, Langmuir 22, 4065 (2006). 20 D. F. Evans and H. Wennerstrom, The Colloidal Domain. (Wiley-VCH: New York,

1994). 21 W. Gotze and L. Sjogren, Rep. Prog. .Phys. 55, 241 (1992). 22 D. Bonn, S. Tanasc, B. Abou, H. Tanaka, and J. Meunier, Phys. Rev. Lett. 89,

157011 (2002). 23 A. Catenaccio, Y. Daruich, and C. Magallanes, Chem. Phys. Lett. 367, 669 (2003). 24 J. D. F. Ramsay, J. Colloid and Interface Sci. 109, 441 (1986). 25 T. Nicolai and S. Cocard, Langmuir 16, 8189 (2000).

12

10-1 100 101

100

101

102

103

104

0.1 1100

101

102

τ /τ

0

1-(tw/t*w)3/2

-2.58

2 % 2.75 %

τ /τ

0

tw/t*w

τ ~ tw

Figure 1. The characteristic dimensionless time scale ( )0τ τ , of the suspension is

plotted against dimensionless age of the sample ( )w wt t∗ for 2 and 2.75 wt. %

suspension of laponite in water. Line is eq. 7 while the experimental data is taken

from Figure 2 of Schosseler et al.5 For 2 % sample, 0τ =2.9 s and wt∗ =13600 s while for

2.75 % sample 0τ =0.94 s and wt∗ =5500 s. Inset shows the same data and the fit, plotted

against ( )3/ 21 w wt t∗− , in the ergodic regime. A power law with exponent –2.58 uniquely

fits the data.